Rewritten and re-annotated because I don’t like the way they did it.
\[\begin{align*} & \left[ \begin{array}{ccc|c} 2 & 1 & -1 & -10 \\ -1 & 2 & 1 & 3 \\ 1 & 2 & 3 & 13 \end{array} \right] \\ & \left[ \begin{array}{ccc|c} 1 & 3 & 0 & -7 \\ -1 & 2 & 1 & 3 \\ 1 & 2 & 3 & 13 \end{array} \right] && R_1 = R_1+R_2 \\ & \left[ \begin{array}{ccc|c} 1 & 3 & 0 & -7 \\ 4 & -4 & 0 & 4 \\ 1 & 2 & 3 & 13 \end{array} \right] && R_2 = -3R_2+R_3 \\ & \left[ \begin{array}{ccc|c} 1 & 3 & 0 & -7 \\ 1 & -1 & 0 & 1 \\ 1 & 2 & 3 & 13 \end{array} \right] && R2= \frac{1}{4}R_2 \\ & \left[ \begin{array}{ccc|c} 0 & 4 & 0 & -8 \\ 1 & -1 & 0 & 1 \\ 1 & 2 & 3 & 13 \end{array} \right] && R_1 = R_1-R_2 \\ & \left[ \begin{array}{ccc|c} 0 & 1 & 0 & -2 \\ 1 & -1 & 0 & 1 \\ 1 & 2 & 3 & 13 \end{array} \right] && R_1 = \frac{1}{2}R_1 \\ & \left[ \begin{array}{ccc|c} 0 & 1 & 0 & -2 \\ 1 & 0 & 0 & -1 \\ 1 & 2 & 3 & 13 \end{array} \right] && R_2=R_2+R_1 \\ & \left[ \begin{array}{ccc|c} 0 & 1 & 0 & -2 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 3 & 18 \end{array} \right] && R_3=R_3-2R_1-R_2 \\ & \left[ \begin{array}{ccc|c} 0 & 1 & 0 & -2 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 6 \end{array} \right] && R_3 = \frac{1}{3}R_3 \end{align*}\]